How can I move many vertices with few dots?
This is standard subdivision surface modeling. The very general idea is that you create a base mesh that is comprised of very simple geometry, and then add a subdivision surface modifier to increase the amount of detail or mesh density.
In your screenshot, the &quot;few selected dots&quot; is the low poly base mesh. The part that has &quot;so many vertices and faces&quot; is the output from adding the subdivision modifier.

To edit the mesh, you just manipulate the low polygon base mesh, and the high resolution version will conform to your edits.
Here is a fantastic tutorial that covers the basics of subdivision modeling:
https://www.youtube.com/watch?v=xmIIIcCUm8E